On the number of antichains of sets in a finite universe

Properties of intervals in the lattice of antichains of subsets of a universe of finite size are investigated. New objects and quantities in this lattice are defined. Expressions and numerical values are deduced for the number of connected antichains and the number of fully distinguishing antichains. The latter establish a connection with Stirling numbers of the second kind. Decomposition properties of intervals in the lattice of antichains are proven. A new operator allowing partitioning the full lattice in intervals derived from lower dimensional sub-lattices is introduced. Special posets underlying an interval of antichains are defined. The poset allows the derivation of a powerful formula for the size of an interval. This formula allows computing intervals in the six dimensional space. Combinatorial coefficients allowing another decomposition of the full lattice are defined. In some specific cases, related to connected components in graphs, these coefficients can be efficiently computed. This formula allows computing the size of the lattice of order 8 efficiently. This size is the number of Dedekind of order 8, the largest one known so far.Comment: Minor changes, typo's and better abstract, line numbers remove

Intervals of Antichains and Their Decompositions

An antichain of subsets is a set of subsets such that no subset in the antichain is a proper subset of any other subset in the antichain. The Dedekind number counts the total number of antichains of subsets of an n-element set. This paper investigates the interval structure of the lattice of antichains. Several partitioning theorems and counting formulas for the size of intervals are derived.Comment: 31page

Decomposition of Intervals in the Space of Anti-Monotonic Functions

Abstract. With the term ’anti-monotonic function’, we designate specific boolean functions on subsets of a finite set of positive integers which we call the universe. Through the well-known bijective relationship between the set of monotonic functions and the set of anti-monotonic functions, the study of the anti-monotonic functions is equivalent to the study of monotonic functions. The true-set of an anti-monotonic function is an antichain. If the universe is denoted by N, the set of anti-monotonic functions is denoted by AMF (N). This set can be partially ordered in a natural way. This paper studies enumeration in the resulting lattice of anti-monotonic functions. We define intervals of anti-monotonic functions according to this order and present four properties of such intervals, Finally we give a formula for the size of a general interval and a recursion formula for the n-th number of Dedekind